87

0

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t (s)

(c)

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t (s)

(d)

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t (s)

(a)

Section 3.5The Area Under the Velocity-Time Graph

3.5The Area Under the Velocity-Time Graph

A natural question to ask about falling objects is how fast they fall, but

Galileo showed that the question has no answer. The physical law that he

discovered connects a cause (the attraction of the planet Earth’s mass) to an

effect, but the effect is predicted in terms of an acceleration rather than a

velocity. In fact, no physical law predicts a definite velocity as a result of a

specific phenomenon, because velocity cannot be measured in absolute

terms, and only changes in velocity relate directly to physical phenomena.

The unfortunate thing about this situation is that the definitions of

velocity and acceleration are stated in terms of the tangent-line technique,

which lets you go from x to v to a, but not the other way around. Without a

technique to go backwards from a to v to x, we cannot say anything quanti-

tative, for instance, about the x-t graph of a falling object. Such a technique

does exist, and I used it to make the x-t graphs in all the examples above.

First let’s concentrate on how to get x information out of a v-t graph. In

example (a), an object moves at a speed of 20 m/s for a period of 4.0 s. The

distance covered is

.

x=v

.

t=(20 m/s)

x

(4.0 s)=80 m. Notice that the quanti-

ties being multiplied are the width and the height of the shaded rectangle

— or, strictly speaking, the time represented by its width and the velocity

represented by its height. The distance of

.

x=80 m thus corresponds to the

area of the shaded part of the graph.

The next step in sophistication is an example like (b), where the object

moves at a constant speed of 10 m/s for two seconds, then for two seconds

at a different constant speed of 20 m/s. The shaded region can be split into

a small rectangle on the left, with an area representing

.

x=20 m, and a taller

one on the right, corresponding to another 40 m of motion. The total

distance is thus 60 m, which corresponds to the total area under the graph.

An example like (c) is now just a trivial generalization; there is simply a

large number of skinny rectangular areas to add up. But notice that graph

(c) is quite a good approximation to the smooth curve (d). Even though we

have no formula for the area of a funny shape like (d), we can approximate

its area by dividing it up into smaller areas like rectangles, whose area is

easier to calculate. If someone hands you a graph like (d) and asks you to

find the area under it, the simplest approach is just to count up the little

rectangles on the underlying graph paper, making rough estimates of

fractional rectangles as you go along.

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t (s)

(b)